For Teaching, Philosophy Makes a Difference

  

What’s the connection between philosophy of mathematics and teaching of mathematics? Each influences the other. The teaching of mathematics should affect the philosophy of mathematics, in the sense that philosophy of mathematics must be compatible with the fact that mathematics can be taught. A philosophy that obscures the teachability of mathematics is unacceptable. Platonists and formalists ignore this question. If mathematical objects were an other-worldly, nonhuman reality (Platonism), or symbols and formulas whose meaning is irrelevant (formalism), it would be a mystery how we can teach it or learn it. Its teachability is the heart of the humanist conception of mathematics.

In the other direction, the philosophy of mathematics held by the teacher can’t help but affect her teaching. The student takes in the teacher’s philosophy through her ears and the textbook’s philosophy through her eyes. The devastating effect of formalism on teaching has been described by others. (See Khinchin or Ernest.) I haven’t seen the effect of Platonism on teaching described in print. But at a teachers’ meeting I heard this:

“Teacher thinks she perceives other-worldly mathematics. Student is convinced teacher really does perceive other-worldly mathematics. No way does student believe he’s about to perceive other-worldly mathematics.” Platonism can justify a student’s certainty that it’s impossible for her/him to understand mathematics. Platonism can justify the belief that some people can’t learn math. Elitism in education and Platonism in philosophy naturally fit together. Humanist philosophy, on the other hand, links mathematics with people, with society, with history. It can’t do damage the way formalism and Platonism can. It could even do good. It could narrow the gap between pupil and subject matter. Such a result would depend on many other factors. But if other factors are compatible, adoption by teachers of a humanist philosophy of mathematics could benefit mathematics education.
**

Political conservatism favors an elite over the lower orders. In mathematics teaching, Platonism suggests that the student either can “see” mathematical reality or she/he can’t.

A humanist/social constructivist/social conceptualist/quasi-empiricist/naturalist/maverick philosophy of mathematics pulls mathematics out of the sky and sets it on earth. This fits with left-wing anti-elitism—its historic striving for universal literacy, universal higher education, universal access to knowledge, and culture. If the Platonist view of number is associated with political conservatism, and the humanist view of number with democratic politics, is that a big surprise?

Karl Popper's Three Worlds

More recently he introduced a notion of “World 3”—a world of scientific and artistic knowledge, distinct from the physical world (World 1) and the world of mind or thought (World 2). In “Epistemology without a knowing subject” and “On the theory of the Objective Mind” he uses “World 3” to mean a world of intelligibles: ideas in the objective sense, possible objects of thought, theories and their logical relations, arguments, and problem situations (1974, p. 154). It seems Popper wants to put a Platonic world of Ideal Forms alongside the mental and physical worlds.

Peter Medawar was impressed by Popper’s World 3. “Popper’s new ontology does away with subjectivism in the world of the mind. Human beings, he says, inhabit or interact with three quite distinct worlds; World 1 is the ordinary physical world, or world of physical states; World 2 is the mental world; World 3 is the world of actual or possible objects of thought—the world of concepts, ideas, theories, theorems, arguments, and explanations—the world, let us say, of all artifacts of the mind. The elements of this world interact with each other much like the ordinary objects of the material world; two theories interact and lead to the formulation of a third; Wagner’s music influences Strauss’s and his in turn all music written since. . . . The existence of World 3, inseparably bound up with human language, is the most distinctly human of all our possessions. The third world is not a fiction, Popper insists, but exists ‘in reality.’ It is a product of the human mind but yet is in large measure autonomous. This was the conception I had been looking for: The third world is the greater and more important part of human inheritance. It’s handing on from generation to generation is what above all else distinguishes man from beast.” But the difficulty of explaining the interaction between these worlds is fatal for Popper, as for his predecessors.

Ethnomathematics

  

Mathematical ideas, like artistic ideas or religious ideas, are a universal part of human culture. This forthright claim isn’t made by Ascher, but her book compels me to that conclusion. Mathematics as we know it was invented by the Greeks. But mathematical ideas involving number and space, probability and logic, even graph theory and group theory—these are present in preliterate societies in North and South America, Africa, the South Pacific, and doubtless many other places if anyone bothers to look.

This is not to say that everybody can do mathematics, any more than everybody can play an instrument or succeed in politics. Many people do not have mathematical or musical or political ability. But every society has its music and its politics; so too, it seems, every society has its mathematics.

Some people count by tens, others by twenties. “There is an often-repeated idea that numerals involving cycles based on ten are somehow more logical because of human fingers. The Yuki of California are said to believe that their cycles based on eight are most appropriate for exactly the same reason. The Yuki, however, are referring to the interfinger spaces.” And how about Toba, a language of western South America, in which “the word with value five implies (two plus three), six implies (two times three), and seven implies (two times three) plus one. Then eight implies (two times four), nine implies (two times four) plus one, and ten is (two times four) plus two.”

Professor Ascher knows of three cultures that trace patterns in sand—the Bushoong in Zaire, the Tshokwe in Zaire and Angola, and the Malekula in Vanuatu (islands between Fiji and Australia formerly called the New Hebrides). Sand drawings play a different role in each culture. “Among the Malekula, passage to the Land of the Dead is dependent on figures traced in the sand. Generally the entrance is guarded by a ghost or spider-related ogre who is seated on a rock and challenges those trying to enter. There is a figure in the sand in front of the guardian and, as the ghost of the newly dead person approaches, the guardian erases half the figure. The challenge is to complete the figure which should have been learned during life, and failure results in being eaten. . . . The tales emphasize the need to know one’s figures properly and demonstrate their cultural importance by involving them in the most fundamental of questions—mortality and (survival) beyond death. The figures vary in complexity from simple closed curves to having more than one hundred vertices, some with degrees of l0 or l2.”

In all three cultures, there’s special concern for Eulerian paths—paths that can be traced through every vertex without tracing any edge more than once. (The seven bridges of Königsberg!) All three seem to know that an Eulerian path is possible if and only if there are zero or two vertices of odd degree. The Maori of New Zealand play a game of skill called mu torere. The game is played by two players; the “board” is an eight-pointed star. Each player has four markers—pebbles, or bits of broken china. Prof. Ascher shows that with any number of points except eight, the game would be uninteresting. “Mu torere, with four markers per player on an eight-pointed star, is the most enjoyable version of the game.”

The Caroline Islanders north of New Guinea cross hundreds of miles of empty ocean to Guam or Saipan. “The Caroline navigators do not use any navigational equipment such as our rulers, compasses, and charts; they travel only with what they carry in their minds.” Professor Ascher reminds us that leading anthropologists once taught that preliterate peoples were at an early stage of evolution. (Western society was advanced.) Later it was said that preliterate peoples (“savages”) had an utterly different way of thinking from us. They were prelogical. We were logical.

Nowadays anthropologists say there’s no objective way to rank societies as more or less advanced, higher or lower. Each is uniquely itself.

Professor Ascher’s research is related to ethnomathematics as an educational program. This movement asks schools to respect and use the mathematical skills pupils bring with them—even if they differ from what’s taught in school. By increasing understanding and respect for ethnomathematics, this work may benefit education.

There’s a lesson for the philosophy of mathematics. Mathematics as an abstract deductive system is associated with our culture. But people created mathematical ideas long before there were abstract deductive systems. Perhaps mathematical ideas will be here after abstract deductive systems have had their day and passed on.

How platonism views mathematics

 

“Mathematical objects are real. Their existence is an objective fact, independent of our knowledge of them. Infinite sets, uncountably infinite sets, infinite-dimensional manifolds, space-filling curves—all the denizens of the mathematical zoo—are definite objects, with definite properties. Some of their properties are known, some are unknown. These objects aren’t physical or material. They’re outside space and time. They’re immutable. They’re uncreated. A meaningful statement about one of these objects is true or false, whether we know it or not. Mathematicians are empirical scientists, like botanists. We can’t invent anything; it’s there already. We try to discover.”

Russell and mathematical certainty

 

 “I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure then the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.”

“Mathematics is, I believe,” says Russell, “the chief source of the belief in eternal and exact truth, as well as in a super-sensible intelligible world. Geometry deals with exact circles, but no sensible object is exactly circular; however carefully we may use our compasses, there will be some imperfections and irregularities. This suggests the view that all exact reasoning applies to ideal as opposed to sensible objects; it is natural to go further, and to argue that thought is nobler than sense, and the objects of thought more real than those of sense-perception. Mystical doctrines as to the relation of time to eternity are also reinforced by pure mathematics, for mathematical objects, such as number, if real at all, are eternal and not in time. Such eternal objects can be conceived as God’s thoughts. Hence Plato’s doctrine that God is a geometer, and Sir James Jeans’ belief that He is addicted to arithmetic. Rationalistic as opposed to apocalyptic religion has been, ever since Pythagoras, and notably ever since Plato, very completely dominated by mathematics and mathematical method.

Foundation for matematics: geometry or algebra?

Until the nineteenth century, geometry was regarded by everybody, including mathematicians, as the most reliable branch of knowledge. Analysis got its meaning and its legitimacy from its link with geometry.

In the nineteenth century, two disasters befell. One was the recognition that there’s more than one thinkable geometry. This was a consequence of the discovery of non-Euclidean geometries.

A second disaster was the overtaking of geometrical intuition by analysis. Space-filling curves** and continuous nowhere-differentiable curves** were shocking surprises. They exposed the fallibility of the geometric intuition on which mathematics rested.

The situation was intolerable. Geometry served from the time of Plato as proof that certainty is possible in human knowledge—including religious certainty. Descartes and Spinoza followed the geometrical style in establishing the existence of God. Loss of certainty in geometry threatened loss of all certainty.

Mathematicians of the nineteenth century rose to the challenge. Led by Dedekind and Weierstrass, they replaced geometry with arithmetic as a foundation for mathematics. This required constructing the continuum—the unbroken line segment—from the natural numbers. Dedekind,** Cantor, and Weierstrass found ways to do this. It turned out that no matter how it was done, building the continuum out of the natural numbers required new mathematical entities—infinite sets.

Descartes (3)

 

The two aspects of Descartes split apart in his intellectual heritage. His Geometry instructed Newton and Leibniz, and is forever integrated into the living body of mathematics. Mathematicians remember him with the name “Cartesian product” for ordered pairs in set theory and geometry.

In theology, the story is more complex. It would be unfair to suppose Descartes’ bows to the Church were insincere. He really was a devout Catholic. His commitment to philosophy and science originated in a dream-vision of the Blessed Virgin.

Descartes was very considerate of the Church’s worries. Still some denounced him as a skeptic or crypto-skeptic. His First Meditation raises profound doubt. Does the Third Meditation really dispel it? Will his “clear and distinct idea” really revive faith, once his doubt has shaken it? In 1663 he was put on the Index (Vrooman, p. 252). In 1679 Leibniz wrote of Descartes’s philosophy, “I do not hesitate to say absolutely that it leads to atheism” (Leibniz, p. 1).

In the following century, nevertheless, Cartesianism became popular among Church apologists. But Descartes’s follower, the “God-intoxicated” Spinoza, was denounced as an atheist, and “Spinozism” became a synonym for atheism. Cartesians were prominent denouncers of Spinoza (Balz, pp. 218–41).

Descartes (2)

 

In Rules for the Direction of the Mind, Descartes wrote: “The first principles themselves are given by intuition alone, while, on the contrary, the remote conclusions are furnished only by deduction. . . . These two methods are the most certain routes to knowledge, and the mind should admit no others. All the rest should be rejected as suspect of error and dangerous.” 

Descartes was embracing the Euclidean ideal: Start from self-evident axioms, proceed by infallible deductions. But in his own research, Descartes forgot the Euclidean ideal. Nowhere in the Geometry do we find the label Axiom, Theorem, or Proof. 

In classical Greece, and again in the Renaissance and after, mathematicians distinguished two ways of proceeding—the “synthetic” and the “analytic.” The synthetic way was Euclid’s: from axioms through deductions to theorems. In the analytic mode, you start with a problem and “analyze” it to find a solution. Today we might call this a “heuristic” or “problem-solving” approach. 

In formal presentation of academic mathematics, the synthetic was and still is the norm. Foundationist schools of the nineteenth and twentieth centuries identify mathematics with its synthetic mode—true axioms followed by correct deductions to yield guaranteed true conclusions. 

In his Rules for the Direction of the Mind, Descartes insists on the synthetic method. But his own research, in the Geometry, uses only the analytic mode. He solves problems. He finds efficient methods for solving problems. Never does he bother with axioms. 

Descartes’s conviction of the certainty of mathematics might lead readers to expect that at least Descartes’s own mathematics is error-free. But of course, as we will see, the Geometry, like every other math book, has mistakes. Certitude is only a goal.

**

You won’t find in the Geometry the method we teach nowadays as Cartesian or “analytic” geometry. Our analytic geometry is based on rectangular coordinates (which we call “Cartesian”). To every point in the plane we associate a pair of real numbers, the “x” and “y” coordinates of the point. To an equation relating x and y corresponds a “graph”—the set of points whose x and y coordinates satisfy the equation. For an equation of first degree, the graph is a straight line. For an equation of second degree, it’s a circle or other conic section. Our idea is to solve geometric problems by reducing them to algebra. Nowhere in Descartes’s book do we see these familiar horizontal and vertical axes! Boyer says it was Newton who first used orthogonal coordinate axes in analytic geometry.

**
The conceptual essence of analytic geometry, the “isomorphism” or exact translation between algebra and geometry, was understood more clearly by Fermat than by Descartes. Fermat’s analytic geometry predated Descartes’s, but it wasn’t published until 1679. The modern formulation comes from a long development. Fermat and Descartes were the first steps. Instead of systematically developing the technique of orthogonal coordinate axes, the Geometry studies a group of problems centering around a problem of Pappus of Alexandria (third century A.D.). To solve Pappus’s problem Descartes develops an algebraic-geometric procedure. First he derives an algebraic equation relating known and unknown lengths in the problem. But he doesn’t then look for an algebraic or numerical solution, as we would do. He is faithful to the Greek conception, that by a solution to a geometric problem is meant a construction with specified instruments. When possible Descartes uses the Euclidean straight edge and compass. When necessary, he brings in his own instrument, an apparatus of hinged rulers. Algebra is an intermediate device, in going from geometric problem to geometric solution. Its role is to reduce a complicated curve to a simpler one whose construction is known. He solves third and fourth-degree equations by reducing them to second degree—to conic sections. He solves certain fifth-and sixth-degree equations by reducing them to third degree. A modern reader knows that the general equation of fifth degree can’t be solved by extraction of roots. So he’s skeptical about Descartes’s claim that his hinged rulers can solve equations of degree six and higher. Descartes was mistaken on several points. In themselves, these are of little interest today. But they discredit his claim of absolute certainty. Descartes’s mathematics refutes his epistemology. 

Emily Grosholz and Carl Boyer point out errors in the Geometry. “When he turns his attention to the locus of five lines, he considers only a few cases, not bothering to complete the task, because, as he says, his method furnishes a way to describe them. But Descartes could not have completed the task, which amounted to giving a catalogue of the cubics. . . . Newton, because he was able to move with confidence between graph and equation, first attempted a catalogue of the cubics; he distinguished seventy-two species of cubics, and even then omitted six” (Grosholz, referring to Whiteside).

**

Descartes claimed his Method was infallible in science and mathematics. He was more cautious with religion. He didn’t derive Holy Scripture or divine revelation by self-evident axioms and infallible deductions. When he heard that Galileo’s Dialogue on the Two Chief Systems was condemned by the Holy Church, he suppressed his first book, Le Monde, even though he was living in Holland, safe from the Church. (Galileo was kept under house arrest at first. For three years he had to recite the seven penitential psalms every week.) Descartes wrote to Father Mersenne, “I would not want for anything in the world to be the author of a work where there was the slightest word of which the Church might disapprove.”

**

Like Pascal, Newton, and Leibniz, Descartes may have valued his contributions to theology above his mathematics. His struggle against skeptics and heretics is the major half of his philosophy, more explicit than his battles with scholastics. “In Descartes’ reply to the objections of Father Bourdin, he announced that he was the first of all men to overthrow the doubts of the Sceptics . . . he discovered how the best minds of the day either spent their time advocating scepticism, or accepted only probable and possibly uncertain views, instead of seeking absolute truth. . . . It was in the light of this awakening to the sceptical menace, that when he was in Paris Descartes set in motion his philosophical revolution by discovering something so certain and so assured that all the most extravagant suppositions brought forward by the sceptics were incapable of shaking . . . in the tradition of the greatest medieval minds, (he) sought to secure man’s natural knowledge to the strongest possible foundation, the all-powerful eternal God” (Popkin, p. 72). The essence of the Meditations is a proof that the world exists by first proving Descartes exists, and then, by contemplating Descartes’s thoughts, proving that God exists and is not a deceiver. Once a non-deceiving God exists, everything else is easy.

 

Descartes

 

D’Alembert wrote that it was Descartes who first “dared . . . to show intelligent minds how to throw off the yoke of scholasticism, of opinion, of authority—in a word, of prejudices and barbarism. . . . He can be thought of as a leader of conspirators who, before anyone else, had the courage to arise against a despotic and arbitrary power, and who, in preparing a resounding revolution, laid the foundations of a more just and happier government which he himself was not able to see established.” 

Philosophers of the scholastic persuasion pointed to the dangerous parallel between Descartes’s scientific individualism and the outlawed Protestant heresy. Descartes said individual thinkers could find scientific truth; Protestants said individual souls could find direct communion with the Almighty. But the Holy Roman Catholic Church knew that individual souls and thinkers could be deceived. It took the experience and wisdom of the Church to prevent the seeker from wandering astray. Despite such scholastic criticism, Descartes quickly came to dominate West European intellectual life.

**

Like Galileo, Descartes recognized mathematics as the principal tool for revealing truths of nature. He was more explicit than Galileo about how to do it. In every scientific problem, said Descartes, find an algebraic equation relating an unknown variable to a known one. Then solve the algebraic equation! With the development of calculus, Descartes’s doctrine was essentially justified. Today we don’t say “find an algebraic equation.” We say “construct a mathematical model.” This is only a technical generalization of Descartes’s idea. Our scientific technology is an inheritance from Descartes.


 

Nicholas of Cusa and Theology in Math

 

The son of a fisherman, Nicholas rose to become a diplomat and counselor for the Church. “He was a member of the commission sent to Constantinople to negotiate with the Eastern church for reunion with Rome, which was temporarily effected at the Council of Florence (1439).” In 1448 he became cardinal and governor of Rome. 

Cusa was not a philosopher of mathematics. He was a philosopher whose thinking was imbued with mathematical images, so that he used mathematics to teach theology. He knew that there are different degrees of infinity. He said, amazingly, that the physical universe is finite but unbounded. He showed that a geometric figure can be both a maximum and a minimum, depending on how it’s parametrized. 

Again from the Encyclopedia, “According to Cusa, a man is wise only if he is aware of the limits of the mind in knowing the truth. . . . Knowledge is learned ignorance (docta ignorantia). Endowed with a natural desire for truth, man seeks it through rational inquiry, which is a movement of the reason from something presupposed as certain to a conclusion that is still in doubt. . . . As a polygon inscribed in a circle increases in number of sides but never becomes a circle, so the mind approximates to truth but never coincides with it. . . . Thus knowledge at best is conjecture (coniectura).” 

Cusa was a Platonist at a time when Aristotelians were dominant. “He constantly criticized the Aristotelians for insisting on the principle of noncontradiction and stubbornly refusing to admit the compatibility of contradictories in reality. It takes almost a miracle, he complained, to get them to admit this; and yet without this admission the ascent of mystical theology is impossible. . . . He constantly strove to see unity and simplicity where the Aristotelians could see only plurality and contradiction. 

“Cusa was most concerned with showing the coincidence of opposites in God. God is the absolute maximum or infinite being, in the sense that he has the fullness of perfection. There is nothing outside him to oppose him or to limit him. He is the all. He is also the maximum, but not in the sense of the supreme degree in a series. As infinite being he does not enter into relation or proportion with finite beings. As the absolute, he excludes all degrees. If we say he is the maximum, we can also say he is the minimum. He is at once all extremes. . . . The coincidence of the maximum and minimum in infinity is illustrated by mathematical figures. For example, imagine a circle with a finite diameter. As the size of the circle is increased, the curvature of the circumference decreases. When the diameter is infinite, the circumference is an absolutely straight line. Thus, in infinity the maximum of straightness is identical with the minimum of curvature. . . . 

Cusa denied that the universe is positively infinite; only God, in his view, could be described in these terms. But he asserted that the universe has no circumference, and consequently that it is boundless or undetermined—a revolutionary notion in cosmology. . . . Just as the universe has no circumference, said Cusa, so it has no fixed center. The earth is not at the center of the universe, nor is it absolutely at rest. Like everything else, it moves in space with a motion that is not absolute but is relative to the observer. . . . 

“Beneath the oppositions and contradictions of Christianity and other religions, he believed there is a fundamental unity and harmony, which, when it is recognized by all men, will be the basis of universal peace.”

Plato's mathematics

  

Plato didn’t have a “philosophy of mathematics” as we understand that phrase today. Mathematics is central in his philosophy. His believes the physical world of visible, changeable entities is illusion. What’s real is invisible, immaterial, eternal. Mathematics is real because it’s immaterial and eternal. It’s tied to religion, as a stepping stone in one’s ascent toward “the good,” the loftiest aspect of invisible reality. A challenge to his notion of mathematics would be a challenge to his religion.

Rules for educating "guardians"

  

My dear Glaucon, what study could draw the soul from the world of becoming to the world of being? . . . this, which they all have in common, which is used in addition by all arts and all sciences and ways of thinking, which is one of the first things every man must learn of necessity.” 

“What’s that?” he asked again. 

“Just this trifle, I said—to distinguish between one and two and three: I mean, in short, number and calculation . . .” 

“Number, then, appears to lead towards the truth?” 

“That is abundantly clear.” 

“Then, as it seems, this would be one of the studies we seek; for this is necessary for the soldier to learn because of arranging his troops, and for the philosopher, because he must rise up out of the world of becoming and lay hold of real being or he will never become a reckoner.” 

“That is true,” said he. 

“Again, our guardian is really both soldier and philosopher.” 

“Certainly.”

“Then, my dear Glaucon, it is proper to lay down that study by law, and to persuade those who are to share in the highest things in the city to go for and tackle the art of calculation, and not as amateurs; they must keep hold of it until they are led to contemplate the very nature of numbers by thought alone, practicing it not for the purpose of buying and selling like merchants or hucksters, but for war, and for the soul itself, to make easier the change from the world of becoming to real being and truth.” 

“Excellently said,” he answered. 

“And besides,” I said, “it comes into my mind, now the study of calculations has been mentioned, how refined that is and useful to us in many ways for what we want, if it is followed for the sake of knowledge and not for chaffering.” 

“How so?” he asked. 

“In this way, as we said just now; how it leads the soul forcibly into some upper region and compels it to debate about numbers in themselves; it nowhere accepts any account of numbers as having tacked onto them bodies which can be seen or touched. . . . I think they are speaking of what can only be conceived in the mind, which it is impossible to deal with in any other way.” 

“You see then, my friend, said I, that really this seems to be the study we need, since it clearly compels the soul to use pure reason in order to find out the truth.” 

“So it most certainly does. . . . 

“For all these reasons, the best natures must be trained in it.” 

After arithmetic, Glaucon and Socrates consider geometry. 

Says Socrates, “The knowledge the geometricians seek is not knowledge of something which comes into being and passes, but knowledge of what always is.”

“Agreed with all my heart, said he, for geometrical knowledge is of that which always is.” 

“A generous admission! Then it would attract the soul toward truth, and work out the philosopher’s mind so as to direct upwards what we now improperly keep downwards.” 

After arithmetic and plane geometry, Glaucon proposes astronomy as the third subject in the curriculum of the Guardians. Socrates objects; solid geometry is more appropriate, he says. 

“Quite so,” says Glaucon, “but it seems that those problems have not yet been solved.” 

“For two reasons,” I said, “because no city holds them in honour, they are weakly pursued, being difficult. Again, the seekers lack a guide, without whom they could not discover; it is hard to find one in the first place, and if they could, as things now are, the seekers in these matters would be too conceited to obey him. But if any whole city should hold these things honourable and take a united lead and supervise, they would obey, and solutions sought constantly and earnestly would become clear. Indeed even now, although dishonoured by the multitude, and held back by the seekers themselves having no conception of the objects for which they are useful, these things do nevertheless force on and grow against all this by their own charm, and I should not be too surprised if they should really come to light. . . . 

“Let us put astronomy as the fourth study, assuming that solid geometry, which we leave aside now, is there for us if only the city would support it.”

[The Republic (“Great Dialogues of Plato,” pp. 315–16, 323–31). Plato gave Socrates the first person, “I.”]

Pythagoreans!

 

The number one, they argued, is the generator of numbers and the number of reason; the number two is the first even or female number, the number of opinion; three is the first true male number, the number of harmony, being composed of unity and diversity; four is the number of justice or retribution, indicating the squaring of accounts; five is the number of marriage, the union of the first true male and female numbers; and six is the number of creation. Each number had its peculiar attributes. The holiest of all was the number ten, or the tetractys, for it represented the number of the universe, including the sum of all possible dimensions. [See also Heath, 1981, p. 75.] A single point is the generator of dimensions, two points determine a line of dimension one, three points S (not on a line) determine a triangle with area of dimension two, and four points (not in a plane) determine a tetrahedron with volume of dimension three; the sum of the numbers representing all dimensions, therefore, is . . . ten. It is a tribute to the abstraction of Pythagorean mathematics that the veneration of the number ten evidently was not dictated by anatomy of the human hand or foot.”

Mainstream Philosophies of Mathematics

 

Frege

                                                          

The name “foundationism” was invented by a prolific name-giver, Imre Lakatos. It refers to Gottlob Frege in his prime, Bertrand Russell in his full logicist phase, Luitjens Brouwer, guru of intuitionism, and David Hilbert, prime advocate of formalism. Lakatos saw that despite their disagreements, they all were hooked on the same delusion: Mathematics must have a firm foundation. They differ on what the foundation should be. 

Foundationism has ancient roots. Behind Frege, Hilbert, and Brouwer stands Immanuel Kant. Behind Kant, Gottfried Leibniz. Behind Leibniz, Baruch Spinoza, and René Descartes. Behind all of them, Thomas Aquinas, Augustine of Hippo, Plato, and the great grandfather of foundationism—Pythagoras. 

We will find that the roots of foundationism are tangled with religion and theology. In Pythagoras and Plato, this intimacy is public. In Kant, it’s half covered. In Frege, it’s out of sight. Then in Georg Cantor, Bertrand Russell, David Hilbert, and Luitjens Brouwer, it pops up like a jack-in-the-box.

In the twentieth century, we look at Russell, Brouwer, Hilbert, Edmund Husserl, Ludwig Wittgenstein, Kurt Gödel, Rudolph Carnap, Willard V. O. Quine, and a small sample of today’s authors. Philip Kitcher said the philosophy of mathematics is generally supposed to begin with Frege—before Frege there was only “prehistory.” Frege transformed the issues constituting philosophy of mathematics. In that sense earlier philosophy can be called prehistoric. But to understand Frege you must see him as a Kantian. To understand Kant you must see his response to Newton, Leibniz, and Hume. Those three go back to Descartes, and through him to Plato. Plato was a Pythagorean. The thread from Pythagoras to Hilbert and Gödel is unbroken. I aim to tell a connected story from Pythagoras to the present—where foundationism came from, where it left us. 

Instead of going straight through from Pythagoras, I’ve split the story into two parallel streams—the first section is about the “Mainstream.” The second is about the “humanists and mavericks.” 

For the Mainstream, mathematics is superhuman—abstract, ideal, infallible, eternal. So many great names: Pythagoras, Plato, Descartes, Spinoza, Leibniz, Kant, Frege, Russell, Carnap. (For Kant, membership in this group is partial.) 

Humanists see mathematics as a human activity, a human creation. Aristotle was a humanist in that sense, as were Locke, Hume, and Mill. Modern philosophers outside the Russell tradition—mavericks—include Peirce, Dewey, Roy Sellars, Wittgenstein, Popper, Lakatos, Wang, Tymoczko, and Kitcher (a self-styled maverick). There are some interesting authors who aren’t labeled philosophers: psychologist Jean Piaget; anthropologist Leslie White; sociologist David Bloor; chemist Michael Polányi; physicist Mario Bunge; educationists Paul Ernest, Gila Hanna, Anna Sfard; mathematicians Henri Poincaré, Alfréd Rényi, George Pólya, Raymond Wilder, Phil Davis, and Brian Rotman.

As mathematics changes...

 

As mathematics grows and changes, geometry changes. A more current example. Until the mid-twentieth century, the “derivative” or “slope” of a function at a point existed only if at that point the graph of the function was smooth—had a definite direction, and no jumps. Now mathematicians have adopted Laurent Schwartz’s generalized functions. Every function, no matter how rough, has a derivative.

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The meaning of differentiation has changed. Newton and Leibniz’s differentiation operator has become something more general. Our generalized differentiation includes the old differentiation, and it’s much more powerful. 

As mathematics grows and changes, functions and operators change.  

A familiar cliché says that while other sciences throw away old theories, mathematics throws away nothing. But the old mathematics isn’t preserved intact. Mathematics is intensely interconnected and self-interactive. The new is vitally linked to the old. The old is revitalized, enriched, and complexified by interaction with the new.

Constructivist - Formalist - Platonist

The constructivist regards the natural numbers as the fundamental datum of mathematics, which neither requires nor is capable of reduction to a more basic notion, and from which all meaningful mathematics must be constructed. 

The Platonist regards mathematical objects as already existing, once and for all, in some ideal and timeless (or tenseless) sense. We don’t create, we discover what’s already there, including infinites of a complexity yet to be conceived by mind of mathematician. 

The formalist rejects both the restrictions of the constructivist and the theology of the Platonist. All that matters are inference rules by which he transforms one formula to another. Any meaning such formulas have is non-mathematical and beside the point.
 

Proof in Mathematics

 

The role of proof in class isn’t the same as in research. In research, it’s to convince. In class, students are all too easily convinced!

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The view I favor is humanism. To the humanist, mathematics is ours—our tool, our plaything. Proof is complete explanation. Give it when complete explanation is appropriate, rather than incomplete explanation or no explanation. The humanist math teacher looks for enlightening proofs, not necessarily the most general or the shortest. Some proofs don’t explain much. They’re called “tricky,” “pulling a rabbit out of a hat.” Give that kind of proof when you want your students to see a rabbit pulled out of a hat. But in general, give proofs that explain. And if the only proof you can find is unmotivated and tricky, if your students won’t learn much from it, must you do it “to stay honest”? That “honesty” is a figment, a self-imposed burden. Better try to be clear, well-motivated, even inspiring. This attitude disturbs people who think proof is the be-all and end-all of mathematics—who say “a mathematician is someone who proves theorems” and “without proof, there’s no mathematics.” From that viewpoint, a mathematics in which proof is less than absolute is heresy. For the humanist, the purpose of proof, as of all teaching, is understanding. Whether to give a proof as is, elaborate it, or abbreviate it, depends on what he thinks will increase the student’s understanding of concepts, methods, and applications.

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In the general classroom, the motto is: “Proof is a tool in service of teacher and class, not a shackle to restrain them.” In teaching future mathematicians, “Proof is a tool in service of research, not a shackle on the mathematician’s imagination.” Proof can convince, and it can explain. In research, convincing is primary. In high-school or undergraduate class, explaining is primary.

Philosophy of mathematics

 

                                                                                                       https://www.durham.ac.uk/departments/academic/mathematical-sciences/

Philosophy of mathematics should be tested against five kinds of mathematical practice: research, application, teaching, history, computing.

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The need to check philosophy of mathematics against mathematical research doesn’t require explication. Many important philosophers of mathematics were mathematical researchers: Pascal, Descartes, Leibniz, d’Alembert, Hilbert, Brouwer, Poincaré, Rényi, and Bishop come to mind. Applied mathematics isn’t illegitimate or marginal. Advances in mathematics for science and technology often are inseparable from advances in pure mathematics. Examples: Newton on universal gravitation and the infinitesimal calculus; Gauss on electromagnetism, astronomy, and geodesy (the last inspired that beautiful pure subject—differential geometry); Poincare on celestial mechanics; and von Neumann on quantum mechanics, fluid dynamics, computer design, numerical analysis, and nuclear explosions.

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G. H. Hardy “famously” boasted: “I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.” Nevertheless, the Hardy-Weinberg law of genetics is better known than his profound contributions to analytic number theory. What’s worse, cryptology is making number theory applicable. Hardy’s contribution to that pure field may yet be useful. Twenty years after the war, mathematical purism was revived, influenced by the famous French group “Bourbaki.” That period is over. Today it’s difficult to find a mathematician who’ll say an unkind word about applied math.

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On the basis of this reduction, philosophers of mathematics generally limit their attention to set theory, logic, and arithmetic. What does this assumption, that all mathematics is fundamentally set theory, do to Euclid, Archimedes, Newton, Leibniz, and Euler? No one dares to say they were thinking in terms of sets, hundreds of years before the set-theoretic reduction was invented. The only way out (implicit, never explicit) is that their own understanding of what they did must be ignored! We know better than they how to explicate their work! That claim obscures history, and obscures the present, which is rooted in history.

An adequate philosophy of mathematics must be compatible with the history of mathematics. It should be capable of shedding light on that history. Why did the Greeks fail to develop mechanics, along the lines that they developed geometry? Why did mathematics lapse in Italy after Galileo, to leap ahead in England, France, and Germany? Why was non-Euclidean geometry not conceived until the nineteenth century, and then independently rediscovered three times? The philosopher of mathematics who is historically conscious can offer such questions to the historian. But if his philosophy makes these questions invisible, then instead of stimulating the history of mathematics, he stultifies it. Computing is a major part of mathematical practice. The use of computing machines in mathematical proof is controversial. An adequate philosophy of mathematics should shed some light on this controversy.

 

Mathematical Objects

 

What’s the nature of mathematical objects?

The question is made difficult by a centuries-old assumption of Western philosophy: “There are two kinds of things in the world. What isn’t physical is mental; what isn’t mental is physical.”

Mental is individual consciousness. It includes private thoughts—mathematical and philosophical, for example—before they’re communicated to the world and become social—and also perception, fear, desire, despair, hope, and so on.

Physical is taking up space—having weight or energy. It’s flesh and bones, sound waves, X-rays, galaxies.

Frege showed that mathematical objects are neither physical nor mental. He labeled them “abstract objects.” What did he tell us about abstract objects? Only this: They’re neither physical nor mental.

Are there other things besides numbers that aren’t mental or physical?

Yes! Sonatas. Prices. Eviction notices. Declarations of war.

Not mental or physical, but not abstract either!

The U.S. Supreme Court exists. It can condemn you to death!

Is the Court physical? If the Court building were blown up and the justices moved to the Pentagon, the Court would go on. Is it mental? If all nine justices expired in a suicide cult, they’d be replaced.

The Court would go on. The Court isn’t the stones of its building, nor is it anyone’s minds and bodies. Physical and mental embodiment are necessary to it, but they’re not it. It’s a social institution. Mental and physical categories are insufficient to understand it. It’s comprehensible only in the context of American society.

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Mathematics consists of concepts. Not pencil or chalk marks, not physical triangles or physical sets, but concepts, which may be suggested or represented by physical objects.

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In reviewing The Mathematical Experience, the mathematical expositor and journalist Martin Gardner made this objection: When two dinosaurs wandered to the water hole in the Jurassic era and met another pair of dinosaurs happily sloshing, there were four dinosaurs at the water hole, even though no human was present to think, “2 + 2 = 4.” This shows, says Gardner, that 2+ 2 really is 4 in reality, not just in some cultural consciousness. 2 + 2 = 4 is a law of nature, he says, independent of human thought.

To untangle this knot, we must see that “2” plays two linguistic roles. Sometimes it’s an adjective; sometimes it’s a noun.

In “two dinosaurs,” “two” is a collective adjective. “Two dinosaurs plus two dinosaurs equals four dinosaurs” is telling about dinosaurs. If I say “Two discrete, reasonably permanent, non-interacting objects collected with two others makes four such objects,” I’m telling part of what’s meant by discrete, reasonably permanent non-interacting objects. That is a statement in elementary physics.

John Stuart Mill pointed out that with regard to discrete, reasonably permanent non-interacting objects, experience tells us

2 + 2 = 4.

In contrast, “Two is prime but four is composite” is a statement about the pure numbers of elementary arithmetic. Now “two” and “four” are nouns, not adjectives. They stand for pure numbers, which are concepts and objects. They are conceptual objects, shared by everyone who knows elementary arithmetic, described by familiar axioms and theorems.

The collective adjectives or “counting numbers” are finite. There’s a limit to how high anyone will ever count. Yet there isn’t any last counting number. If you counted up to, say, a billion, then you could count to a billion and one. In pure arithmetic, these two properties—finiteness, and not having a last—are contradictory. This shows that the counting numbers aren’t the pure numbers.

Consider the pure number 10^(1010). We easily ascertain some of its properties, such as: “The only prime factors of 10^(1010) are 2 and 5.” But we can’t count that high. In that sense, there’s no counting number equal to 10^(1010).

Körner made the same distinction, using uppercase for Counting Numbers (adjectives) and lowercase for “pure” natural numbers (nouns). Jacob Klein wrote that a related distinction was made by the Greeks, using their words “arithmos” and “logistiké.”

So “two” and “four” have double meanings: as Counting Numbers or as pure numbers. The formula
 

2 + 2 = 4
 

has a double meaning. It’s about counting—about how discrete, reasonably permanent, non-interacting objects behave. And it’s a theorem in pure arithmetic (Peano arithmetic if you like). This linguistic ambiguity blurs the difference between Counting Numbers and pure natural numbers. But it’s convenient. It’s comparable to the ambiguity of non-mathematical words, such as “art” or “America.”

The pure numbers rise out of the Counting Numbers. In a process related to Aristotle’s abstraction, they disconnect from “real” objects, to exist as shared concepts in the mind/brains of people who know elementary arithmetic. In that realm of shared concepts, 2 + 2 = 4 is a different fact, with a different meaning. And we can now show that it follows logically from other shared concepts, which we usually call axioms.

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Once created and communicated, mathematical objects are there. They detach from their originator and become part of human culture. We learn of them as external objects, with known properties and unknown properties. Of the unknown properties, there are some we are able to discover.

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Why do these objects, our own creations, so often become useful in describing nature? To answer this in detail is a major task for the history of mathematics, and for a psychology of mathematical cognition that may be coming to birth in Piaget and Vygotsky. To answer it in general, however, is easy. Mathematics is part of human culture and history, which are rooted in our biological nature and our physical and biological surroundings. Our mathematical ideas in general match our world for the same reason that our lungs match earth’s atmosphere.

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Psychological and historical studies won’t make mathematical truth indubitable. But why expect mathematical truth to be indubitable? Correcting errors by confronting them with experience is the essence of science. What’s needed is explication of what mathematicians do—as part of general human culture, as well as in mathematical terms. The result will be a description of mathematics that mathematicians recognize—the kind of truth that’s obvious once said.  

Certain kinds of ideas (concepts, notions, conceptions, and so forth) have science-like quality. They have the rigidity, the reproducibility, of physical science. They yield reproducible results, independent of particular investigators. Such kinds of ideas are important enough to have a name. Study of the lawful, predictable parts of the physical world has a name: “physics.” 

Study of the lawful, predictable, parts of the social-conceptual world also has a name: “mathematics.” A world of ideas exists, created by human beings, existing in their shared consciousness. These ideas have objective properties, in the same sense that material objects have objective properties. The construction of proof and counterexample is the method of discovering the properties of these ideas. This branch of knowledge is called mathematics.